It is obvious that the integration in (REF ) involves the term \(k u_{ x}(x)\)  which is not \(L^2(\mathbb {R})\)  bounded since \(k\)  may go to infinity.
In our previous work, through the small \(k\)  and large \(k\)  estimates respectively, we overcome this difficulty and prove the existence and the differentiability of the Jost functions
\(\psi ^{\pm }(x,t;k)\)  [1]}.
